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The puzzle.
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Back in the 1860's, five sailors came to be marooned on a desert
island after a violent storm at sea. They soon realised that they
were going to need to find food, so they all agreed to go and
collect coconuts, watched by a bewildered monkey sitting at the
top of a palm tree.

At the end of an exhausting day they had a large pile of coconuts
at the bottom of the palm tree, which they agreed to share out the
following morning. The sailors then all went off to sleep.

After a while one of the sailors woke up, and thought to himself
"I expect there will be arguments in the morning when it comes
time to share things out", so he decided to take his share there
and then. He split the pile into five equal shares, and found he
had one left over, which he threw to the monkey. He then took his
fifth and hid it away, recombined the remainder and went back to
sleep.

Sometime later a second sailor woke up and thought the same as the
first. He split the remaining pile into five equal parts, found
he had one left over which he threw to the monkey, took his share
and hid it, and recombined the now diminishing pile.

Over the course of the night all the sailors did the same thing,
and each time they split the pile there was one left over which
was thrown to the monkey.

In the morning the five sailors grouped around the now tiny pile
of coconuts. Each knew what had happened but nobody wanted to say
anything, so they proceeded to divide the pile into five equal
shares, watched by a rather obese monkey! Again they found there
was still one left over which they threw to the monkey.

What is the minimum number of coconuts that the sailors could have
collected and piled up?

The solution is provided below, after a long gap of blank lines. If 
you want to try and work it out yourself may we suggest you don't
scroll down!


































The Solution.
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The smallest number of coconuts possible is 15621. Try it - it does
work!

It is very difficult to calculate this directly, but there is a
simple way of reaching that answer if you realise that for every
answer you find, the next larger answer can be obtained by adding
5 to the power 6, or 15625. (6 equal divisions by 5).

If you search for a small positive answer you will be out of luck,
but how about if there is a small negative number, that while 
impossible in practice actually correctly satisfies the steps as
outlined?

If you try -4 you will find it works perfectly, as long as you throw
the positive coconut to the monkey first!

Let's take the first step. Start with -4 coconuts, throw a positive 
coconut to the monkey, which leaves you a pile of -5 coconuts. Then
split that into 5 piles of -1 coconut each, stash one of the
negative coconuts away somewhere, then recombine the remaining -4
coconuts and you are back where you started, and you can continue 
the process for the six steps required (or indeed ad infinitum!).

Adding 15625 to -4 will then get you to the first positive answer
of 15621.